Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 10 (2014), 072, 10 pages
The GraviGUT Algebra Is not a Subalgebra of E8,
but E8 Does Contain an Extended GraviGUT Algebra
Andrew DOUGLAS
†
and Joe REPKA
‡
†
CUNY Graduate Center and New York City College of Technology,
City University of New York, USA
E-mail: adouglas2@gc.cuny.edu
‡
Department of Mathematics, University of Toronto, Canada
E-mail: repka@math.toronto.edu
Received April 04, 2014, in final form July 03, 2014; Published online July 08, 2014
http://dx.doi.org/10.3842/SIGMA.2014.072
Abstract. The (real) GraviGUT algebra is an extension of the spin(11, 3) algebra by a 64dimensional Lie algebra, but there is some ambiguity in the literature about its definition.
Recently, Lisi constructed an embedding of the GraviGUT algebra into the quaternionic real
form of E8 . We clarify the definition, showing that there is only one possibility, and then
prove that the GraviGUT algebra cannot be embedded into any real form of E8 . We then
modify Lisi’s construction to create true Lie algebra embeddings of the extended GraviGUT
algebra into E8 . We classify these embeddings up to inner automorphism.
Key words: exceptional Lie algebra E8 ; GraviGUT algebra; extended GraviGUT algebra;
Lie algebra embeddings
2010 Mathematics Subject Classification: 17B05; 17B10; 17B20; 17B25; 17B81
1
Introduction
The Standard Model of particle physics, with gauge group U(1) × SU(2) × SU(3), attempts to
describe all particles and all forces, except gravity. Grand Unified Theories (GUT) attempt to
unify the forces and particles of the Standard Model. The three main GUTs are Georgi and
Glashow’s SU(5) theory, Georgi’s Spin(10) theory, and the Pati–Salam model based on the Lie
group SU(2) × SU(2) × SU(4) [2].
In [8], Lisi attempts to construct a unification which includes gravity. In this construction,
Lisi first embeds gravity and the standard model into spin(11, 3). He then embeds spin(11, 3)
together with the positive chirality 64-dimensional spin(11, 3) irrep into the quaternionic real
form of E8 . Lisi refers to the embedded Lie algebra as the GraviGUT algebra. For a description
of Lisi’s theory see [8] or [7]. For a critique of Lisi’s theory involving the GraviGUT algebra
see [3]. We note that the GraviGUT algebra was first introduced by Nesti and Percacci [12].
In Section 4 of [11], it is observed that one of the Lie algebras associated with a point in
Vogel’s plane (cf. [13]) has the same dimension as the GraviGUT algebra, and it is hypothesized
there that these two algebras are in fact isomorphic. It would certainly be interesting to identify
the algebra in question.
Because the exposition of [8] reflects the process of exploring the possible realizations of the
GraviGUT algebra, there is some potential for confusion about its definition. Lisi first describes
the algebraic structure of spin(11, 3) and the action of spin(11, 3) on its 64-dimensional irrep V
in equations (3.9) and (3.10) from [8], respectively. He then notes that the structure of the
GraviGUT algebra could be completed by defining a trivial Lie bracket on V .
However, the actual algebraic structure of his explicit realization of V is not established until
equations (4.3) and (4.4) from [8], when Lisi describes his embedding; it is here that we see
2
A. Douglas and J. Repka
that V is not abelian relative to the Lie bracket inherited from E8 . In fact, with this definition
of the Lie bracket, the subspace of E8 spanned by the embedded copies of spin(11, 3) and V is
not a Lie algebra at all: it is not closed under the bracket. Lisi acknowledges this in his remark
near the end of § 3 that “The word ‘algebra’ is used here in a generalized sense”.
Theorem 2 of the present paper actually shows that the only way to extend the usual bracket
on so(14)C and its action on V to make so(14)C A V into a Lie algebra is to require V to be
abelian. In particular, the only possible definition of the (complexified) GraviGUT algebra as
a Lie algebra is so(14)C A V , where V is a 64-dimensional abelian ideal which is irreducible
under the action of so(14)C .
Once the structure of the complexified GraviGUT algebra is specified, it is not difficult to
show that it cannot be embedded into the complex algebra E8 ; cf. Corollary 1. Hence, the (real)
GraviGUT algebra cannot be embedded into the quaternionic real form of E8 , or any other real
form of E8 .
However, the operators in E8 described by Lisi do generate a larger Lie algebra, which contains
an additional 14-dimensional ideal. We call this larger algebra the extended GraviGUT algebra.
We modify Lisi’s construction to create true Lie algebra embeddings of the extended GraviGUT algebra into E8 . The (complex) extended GraviGUT algebra is a nonabelian, nilpotent
extension of so(14)C by a 78-dimensional so(14)C -representation. This 78-dimensional representation is composed of a 64-dimensional irrep and the standard 14-dimensional so(14)C -irrep. Its
precise structure is described in Section 6, but we do note here that the (complexified) GraviGUT algebra is a quotient of the extended GraviGUT algebra. We classify these embeddings
up to inner automorphism.
The article is organized as follows. Section 2 contains relevant background on Lie algebras
and their representations: in particular, it deals with the complex, simple Lie algebras so(14)C
and E8 . Section 3 presents additional notation and terminology. In Section 4 we describe the
classification of embeddings of so(14)C into E8 , which will be used in the following section. In
Section 5 we determine the only possible definition of the GraviGUT algebra and also establish
that the complexified GraviGUT algebra cannot be embedded into the complex algebra E8 .
Finally, in Section 6 we classify the embeddings of the extended GraviGUT algebra into E8 .
2
The complex Lie algebras so(14)C and E8 ,
and their representations
The special orthogonal algebra so(14)C is the complexification of spin(11, 3). It is the Lie algebra
of complex 14 × 14 matrices N satisfying N tr = −N . The dimension of so(14)C is 91 and its
rank is 7. The Lie group corresponding to so(14)C arises naturally as the symmetry group of
a projective space over R [1].
E8 is the complex, exceptional Lie algebra of rank 8. It is 248-dimensional. Like so(14)C ,
E8 has a close connection to the Riemannian geometry of projective spaces (for details, we refer
the reader to [1]).
Let g denote so(14)C or E8 . Let k = 7 or 8 when g = so(14)C or E8 , respectively. We may
define g by a set of generators {Hi , Xi , Yi }1≤i≤k together with the Chevalley–Serre relations [6]:
[Hi , Hj ] = 0,
g
[Hi , Xj ] = Mji
Xj ,
g
[Hi , Yj ] = −Mji
Yj ,
g
1−Mji
(ad Xi )
[Xi , Yj ] = δij Hi ,
g
(Xj ) = 0,
(ad Yi )1−Mji (Yj ) = 0,
when
i 6= j.
Here 1 ≤ i, j ≤ k, and M g is the Cartan matrix of g. The Xi , for 1 ≤ i ≤ k, correspond to the
simple roots. We write H for the Cartan subalgebra spanned by {Hi }.
The GraviGUT Algebra Is not a Subalgebra of E8
3
For future reference, the Dynkin diagrams of so(14)C and E8 , indicating the numbering of
simple roots, are given in Fig. 1.
Figure 1. Dynkin diagrams of so(14)C and E8 .
We now briefly describe the finite-dimensional, irreducible representations (irreps) of so(14)C
g
and E8 , with g and k defined as above. For i = 1, . . . , k, define αi , λi ∈ H ∗ by αi (Hj ) = Mji
,
g
and λi (Hj ) = δij , where M is the Cartan matrix of g. The λi are the fundamental weights, and
their indexing corresponds with that of the Dynkin diagram of type so(14)C or E8 in Fig. 1.
For each λ = m1 λ1 + · · · + mk λk ∈ H ∗ with nonnegative integers m1 , . . . , mk , there exists
an irrep of g with highest weight λ, denoted Vg (λ). The irreps Vg (λi ) for 1 ≤ i ≤ k are the
fundamental representations. Each irrep of g is equivalent to Vg (λ), where λ = m1 λ1 +· · ·+mk λk
for some nonnegative integers m1 , . . . , mk .
3
Additional def initions and notation
The following definitions and notation will be used in this article:
• For 1 ≤ ai ≤ 8, let Xai correspond to the ai th simple root of E8 . We then define
Xa1 ,a2 ,a3 ,...,am ≡ [[. . . [[Xa1 , Xa2 ], Xa3 ], . . .], Xam ].
Ya1 ,a2 ,a3 ,...,am is defined analogously.
• Let ϕ : so(14)C ,→ E8 be an embedding. Further, let W be an element of E8 . Then,
[W ]ϕ(so(14)C ) is the so(14)C representation generated by W with respect to the adjoint
action of ϕ(so(14)C ). When the embedding ϕ is clear, as will be the case below, we simply
write [W ]so(14)C .
• Let ϕ and % be Lie algebra embeddings of g0 into g. Then ϕ and % are equivalent if there
is an inner automorphism ρ : g → g such that ϕ = ρ ◦ %, and we write
ϕ ∼ %.
• Two embeddings ϕ and % of g0 into g are linearly equivalent if for each representation
π : g → gl(V ) the induced g0 representations π ◦ ϕ, π ◦ % are equivalent, and we write
ϕ ∼L %.
Clearly equivalence of embeddings implies linear equivalence, but the converse is not in
general true.
We define equivalence and linear equivalence of subalgebras much as we did for embeddings:
• Two subalgebras g0 and g00 of g are equivalent if there is an inner automorphism ρ of g
such that ρ(g0 ) = g00 .
• Two subalgebras g0 and g00 of g are linearly equivalent if for every representation π : g →
gl(V ) the subalgebras π(g0 ), π(g00 ) of gl(V ) are conjugate under GL(V ).
4
4
A. Douglas and J. Repka
Embedding so(14)C into E8
In [4], the authors presented the following well-known “natural” embedding of so(14)C into E8 :
ϕ : so(14)C ,→ E8 ,
H8−i 7→ Hi+1 ,
X8−i 7→ Xi+1 ,
Y8−i 7→ Yi+1 ,
(1)
where 1 ≤ i ≤ 7. This embedding may be visualized as a “natural” subgraph of the Dynkin
diagram of E8 which is isomorphic to the Dynkin diagram of so(14)C (see Fig. 1).
In [10], Minchenko showed that there is a unique subalgebra isomorphic to so(14)C in E8 , up
to inner automorphism. Hence, the only way to get new embeddings of so(14)C into E8 other
than the ϕ described in equation (1) is to compose ϕ with an outer automorphism of so(14)C .
However, it was shown by the authors in [4] that outer automorphisms of so(14)C do not produce
new embeddings of so(14)C into E8 . We thus have the following theorem [4].
Theorem 1. The map ϕ : so(14)C ,→ E8 defined in equation (1) is the unique embedding of
so(14)C into E8 , up to inner automorphism.
5
The GraviGUT algebra is not a subalgebra of E8
Theorem 2. Consider a sum of complex vector spaces so(14)C ⊕V , where V is a 64-dimensional
space. Suppose a Lie bracket is defined which gives the usual structure to the so(14)C subspace
and in such a way that the brackets [X, v], for X ∈ so(14)C and v ∈ V , define an action
of so(14)C on V under which V becomes an irreducible so(14)C -module. Then the only way to
extend this bracket to make so(14)C A V into a Lie algebra is to put the abelian structure on V ,
i.e., [v, v 0 ] = 0, for all v, v 0 ∈ V .
In particular, the only possible definition of the (complexified) GraviGUT algebra as a Lie
algebra is so(14)C A V , where V is a 64-dimensional abelian ideal which is irreducible under the
action of so(14)C .
Proof . Let V be a 64-dimensional so(14)C -irrep. Then V ∼
= V (λ7 ). Consider
= V (λ6 ) or V ∼
the tensor product decompositions:
V (λ6 ) ⊗ V (λ6 ) ∼
= V (2λ6 ) ⊕ V (λ5 ) ⊕ V (λ3 ) ⊕ V (λ1 ),
V (λ7 ) ⊗ V (λ7 ) ∼
= V (2λ7 ) ⊕ V (λ5 ) ⊕ V (λ3 ) ⊕ V (λ1 ).
Since V (λ6 ) does not occur in the tensor product decomposition of V (λ6 ) ⊗ V (λ6 ), V (λ7 ) does
not occur in the tensor product decomposition of V (λ7 ) ⊗ V (λ7 ), and neither decomposition
contains a 91-dimensional irrep, we cannot have a nontrivial product V ⊗ V → V or V ⊗ V →
so(14)C . Hence, we cannot have a nontrivial product V × V → V or V × V → so(14)C or
V × V → (so(14)C A V ). The only possible definition of a Lie algebra structure is to make V
an abelian subalgebra.
Corollary 1. The GraviGUT algebra cannot be embedded into the quaternionic real form of E8 ,
or any other real form of E8 .
Proof . The maximal dimension of an abelian subalgebra of E8 is 36 [9]. This implies that V
cannot be a subalgebra of E8 , and that the complexified GraviGUT algebra cannot be embedded
into E8 . The result follows.
The GraviGUT Algebra Is not a Subalgebra of E8
5
Remarks.
1. We also note that if we have a 64-dimensional representation V that is not irreducible,
then embedding so(14)C A V into E8 is still not possible. The summands in the direct
sum decomposition of E8 as an ϕ(so(14)C )-module have dimensions 1, 14, 14, 64, 64, 91,
as we will see below in equation (2). Hence, the only 64-dimensional so(14)C submodules
of E8 are irreducible.
2. The subspace of E8 defined by Lisi in his equations (4.3), (4.4) (see [8]) is not closed under
the Lie bracket it inherits from E8 .
6
The extended GraviGUT algebra in E8
In [4], the authors computed the following decomposition of E8 with respect to the adjoint action
of ϕ(so(14)C ):
E8 ∼
(2)
=so(14)C V(λ2 ) ⊕ V(λ1 ) ⊕ V(λ6 ) ⊕ V(λ7 ) ⊕ V(λ1 ) ⊕ V(0)
∼
=so(14)C [X74 ]so(14)C ⊕ [X120 ]so(14)C ⊕ [Y1 ]so(14)C ⊕ [X112 ]so(14)C ⊕ [Y97 ]so(14)C ⊕ [H]so(14)C ,
where
X74 = X4,5,6,7,8,2,3,4,5,6,7 ,
X112 = −X3,4,2,1,5,4,3,6,5,4,7,2,6,5,8,7,6,4,5,3,4,2 ,
X120 = X8,7,6,5,4,3,2,1,4,5,6,7,3,4,5,6,2,4,5,3,4,2,1,3,4,5,6,7,8 ,
Y97 = −Y5,4,2,3,6,4,1,3,5,4,7,2,6,5,4,3,1 ,
H = 4H1 + 5H2 + 7H3 + 10H4 + 8H5 + 6H6 + 4H7 + 2H8 .
Lemma 1. The following are 78-dimensional, nonabelian nilpotent subalgebras of E8 :
[Y1 ]so(14)C ⊕ [Y97 ]so(14)C ,
[X112 ]so(14)C ⊕ [X120 ]so(14)C .
Note that the sums are direct as so(14)C irreps, but not as subalgebras of E8 . Further, the
subalgebras [Y97 ]so(14)C and [X120 ]so(14)C of E8 are abelian.
Proof . The authors showed in [4] that [Y97 ]so(14)C and [X120 ]so(14)C are abelian subalgebras
of E8 .
The positive roots of E8 , as explicitly described in Appendix A, give us a triangular decomposition of E8 : E8,+ ⊕ E8,0 ⊕ E8,− . In Appendix A we also explicitly describe bases for
the representations [X120 ]so(14)C , [X112 ]so(14)C , [Y97 ]so(14)C , and [Y1 ]so(14)C . Since ϕ(so(14)C ) =
[X74 ]so(14)C , we also give bases for ϕ(so(14)C+ ) and ϕ(so(14)C− ). Each of these bases consists of
all positive root vectors, or all negative root vectors.
The bases given in Appendix A imply
E8,+ = ϕ(so(14)C+ ) ⊕ [X112 ]so(14)C ⊕ [X120 ]so(14)C ,
E8,− = ϕ(so(14)C− ) ⊕ [Y1 ]so(14)C ⊕ [Y97 ]so(14)C .
And of course
[[X112 ]so(14)C ⊕ [X120 ]so(14)C , [X112 ]so(14)C ⊕ [X120 ]so(14)C ],
[[Y1 ]so(14)C ⊕ [Y97 ]so(14)C , [Y1 ]so(14)C ⊕ [Y97 ]so(14)C ]
are subsets of E8,+ and E8,− , respectively.
6
A. Douglas and J. Repka
In Appendix A, we see that the positive root vector Xα is in the basis of [X112 ]so(14)C or
[X120 ]so(14)C if α1 6= 0, where α1 is the first entry of α. If α1 = 0, then Xα is in the basis of
ϕ(so(14)C+ ).
Thus, if Xα and Xα0 are positive root vectors in the basis of [X112 ]so(14)C or [X120 ]so(14)C such
that [Xα , Xα0 ] 6= 0, then this product is a nonzero scalar multiple of Xα+α0 , where (α + α0 )1 6= 0,
so that Xα+α0 is an element of [X112 ]so(14)C ⊕ [X120 ]so(14)C . Therefore,
[[X112 ]so(14)C ⊕ [X120 ]so(14)C , [X112 ]so(14)C ⊕ [X120 ]so(14)C ] ⊆ [X112 ]so(14)C ⊕ [X120 ]so(14)C .
In a similar manner we show
[[Y1 ]so(14)C ⊕ [Y97 ]so(14)C , [Y1 ]so(14)C ⊕ [Y97 ]so(14)C ] ⊆ [Y1 ]so(14)C ⊕ [Y97 ]so(14)C .
Thus [Y1 ]so(14)C ⊕ [Y97 ]so(14)C and [X112 ]so(14)C ⊕ [X120 ]so(14)C are subalgebras of E8 . Further, they
are nilpotent since they are contained in E8,− or E8,+ , respectively.
Lemma 2. The following are not subalgebras of E8 :
[Y97 ]so(14)C ⊕ [X112 ]so(14)C ,
[X120 ]so(14)C ⊕ [Y1 ]so(14)C .
Proof . Referring to the bases of [X120 ]so(14)C and [Y1 ]so(14)C described in Appendix A, we have
Y112 ∈ [Y1 ]so(14)C , and of course X120 ∈ [X120 ]so(14)C . However, [Y112 , X120 ] is a nonzero multiple of X47 , which is not in [X120 ]so(14)C ⊕ [Y1 ]so(14)C . Hence [X120 ]so(14)C ⊕ [Y1 ]so(14)C is not
a subalgebra. Similarly [Y97 ]so(14)C ⊕ [X112 ]so(14)C is not a subalgebra.
We may now explicitly define the extended GraviGUT algebra as follows. As a vector space,
it is
so(14)C A (V (λ6 ) ⊕ V (λ1 )) ∼
= so(14)C A (V (λ7 ) ⊕ V (λ1 )).
The Lie algebra structure is inherited from that of E8 . In particular, the following subalgebras
are not direct sums as algebras, though the sums are direct as vector spaces:
V (λ6 ) + V (λ1 ) ∼
= [Y1 ]so(14)C + [Y97 ]so(14)C ,
V (λ7 ) + V (λ1 ) ∼
= [X112 ]so(14)C + [X120 ]so(14)C .
We note further that not only are so(14)C A (V (λ6 ) + V (λ1 )) and so(14)C A (V (λ7 ) + V (λ1 ))
isomorphic subalgebras, but they are equivalent subalgebas of E8 , related by the Chevalley
involution of E8 . Hence, we shall only consider so(14)C A (V (λ7 ) + V (λ1 )). It is significant
to observe that the (complexified) GraviGUT algebra is a quotient of the extended GraviGUT
algebra. The only distinction that can be made is that these two subalgebras of E8 are not the
same as so(14)C -modules.
We now proceed to the classification of embeddings of the extended GraviGUT algebra
into E8 . A lift of ϕ : so(14)C ,→ E8 to so(14)C A (V (λ7 )+V (λ1 )) is completely determined by its
definition on highest weight vectors of V (λ7 ) and V (λ1 ). Call these vectors u and v, respectively.
Hence, for any α, β ∈ C∗ , the following is a lift of ϕ : so(14)C ,→ E8 to so(14)C A (V (λ7 )+V (λ1 )):
ϕ
eα,β : so(14)C A (V (λ7 ) + V (λ1 )) ,→ E8 ,
u 7→ αX112 ,
v 7→ βX120 .
(3)
Theorem 3. All embeddings of the extended GraviGUT algebra into E8 are given by ϕ
eα,β , for
∗
all α, β ∈ C . These embeddings are classified according to the rule
ϕ
eα,β ∼ ϕ
eα0 ,β 0 ⇔ α02 β = α2 β 0 .
The GraviGUT Algebra Is not a Subalgebra of E8
7
Proof . First note that by Theorem 1, all embeddings of the extended GraviGUT algebras must
come from lifts of ϕ, and hence, considering Lemmas 1 and 2, equation (3) defines all embeddings
of the extended GraviGUT algebra into E8 .
Define inner automorphisms of E8 as follows:
ρ:
X1 7→ αX1 ,
Xi 7→ Xi ,
ρ0 :
X1 7→ α0 X1 ,
Xi 7→ Xi ,
1
Y1 ,
α
Yi 7→ Yi ,
1
Y1 7→ 0 Y1 ,
α
Yi 7→ Yi ,
Y1 7→
for 2 ≤ i ≤ 8. Then ϕ
eα,β = ρ ◦ ϕ
e1,
β
α2
, and ϕ
eα0 ,β 0 = ρ0 ◦ ϕ
e1, β0 . Hence ϕ
eα,β ∼ ϕ
e1,
α02
∼ϕ
e1, β0 .
β
α2
, and ϕ
eα0 ,β 0
α02
If ϑ is an inner automorphism of E8 such that ϑ ◦ ϕ
e1,
β
α2
=ϕ
e1, β0 , then ϑ fixes Xi and Yi for
α02
2 ≤ i ≤ 8, and also X112 . We have
[. . . [X112 , Y2 ], Y4 ], Y3 ], Y5 ], Y6 ], Y7 ], Y8 ], Y4 ], Y5 ],
Y2 ], Y6 ], Y4 ], Y5 ], Y3 ], Y4 ], Y7 ], Y6 ], Y5 ], Y2 ], Y4 ], Y3 ] = X1 ,
so that ϑ fixes X1 . Hence ϑ(X120 ) = X120 , so that
obvious. Hence we have established
β
α2
=
β0
.
α02
The opposite implication is
ϕ
eα,β ∼ ϕ
eα,β ⇔ α02 β = α2 β 0 .
Remark. Theorem 3 implies, of course, that there are an infinite number of embeddings of the
extended GraviGUT algebra into E8 , up to inner automorphism. However, it is interesting to
note that there is a unique subalgebra of E8 which is isomorphic to the extended GraviGUT
algebra up to inner automorphism.
7
Conclusions
In [8], Lisi identified a copy of spin(11, 3) in the quaternionic real form of E8 and a 64-dimensional
subspace on which spin(11, 3) acts irreducibly. His hope was to embed the GraviGUT algebra.
However, the subspace spanned by these spaces is not closed under the Lie bracket.
We proved that the only possible Lie algebra structure on the GraviGUT algebra has a trivial
(abelian) bracket on the 64-dimensional subspace. In particular, the GraviGUT algebra cannot
be embedded into any real form of E8 . We then modified Lisi’s construction to create true Lie
algebra embeddings of the extended GraviGUT algebra into E8 . We classified these embeddings
up to inner automorphism.
A
The representations [X74 ]so(14)C , [X120 ]so(14)C , [X112 ]so(14)C ,
[Y97 ]so(14)C , and [Y1 ]so(14)C
In this appendix we describe the representations [X74 ]so(14)C , [X120 ]so(14)C , [X112 ]so(14)C ,
[Y97 ]so(14)C and [Y1 ]so(14)C from equation (2).
Let α1 , α2 , α3 , . . . , α8 be a set of simple roots for E8 . To any positive root a1 α1 + a2 α2 +
a3 α3 + · · · + a8 α8 we may associate a vector [a1 , a2 , a3 , . . . , a8 ] ∈ Z8≥0 . With this convention, the
positive roots of E8 , as computed with GAP [5], are as follows:
α1 = [1, 0, 0, 0, 0, 0, 0, 0],
α2 = [0, 1, 0, 0, 0, 0, 0, 0],
8
A. Douglas and J. Repka
α3 = [0, 0, 1, 0, 0, 0, 0, 0],
α4 = [0, 0, 0, 1, 0, 0, 0, 0],
α5 = [0, 0, 0, 0, 1, 0, 0, 0],
α6 = [0, 0, 0, 0, 0, 1, 0, 0],
α7 = [0, 0, 0, 0, 0, 0, 1, 0],
α8 = [0, 0, 0, 0, 0, 0, 0, 1],
α9 = [1, 0, 1, 0, 0, 0, 0, 0],
α10 = [0, 1, 0, 1, 0, 0, 0, 0],
α11 = [0, 0, 1, 1, 0, 0, 0, 0],
α12 = [0, 0, 0, 1, 1, 0, 0, 0],
α13 = [0, 0, 0, 0, 1, 1, 0, 0],
α14 = [0, 0, 0, 0, 0, 1, 1, 0],
α15 = [0, 0, 0, 0, 0, 0, 1, 1],
α16 = [1, 0, 1, 1, 0, 0, 0, 0],
α17 = [0, 1, 1, 1, 0, 0, 0, 0],
α18 = [0, 1, 0, 1, 1, 0, 0, 0],
α19 = [0, 0, 1, 1, 1, 0, 0, 0],
α20 = [0, 0, 0, 1, 1, 1, 0, 0],
α21 = [0, 0, 0, 0, 1, 1, 1, 0],
α22 = [0, 0, 0, 0, 0, 1, 1, 1],
α23 = [1, 1, 1, 1, 0, 0, 0, 0],
α24 = [1, 0, 1, 1, 1, 0, 0, 0],
α25 = [0, 1, 1, 1, 1, 0, 0, 0],
α26 = [0, 1, 0, 1, 1, 1, 0, 0],
α27 = [0, 0, 1, 1, 1, 1, 0, 0],
α28 = [0, 0, 0, 1, 1, 1, 1, 0],
α29 = [0, 0, 0, 0, 1, 1, 1, 1],
α30 = [1, 1, 1, 1, 1, 0, 0, 0],
α31 = [1, 0, 1, 1, 1, 1, 0, 0],
α32 = [0, 1, 1, 2, 1, 0, 0, 0],
α33 = [0, 1, 1, 1, 1, 1, 0, 0],
α34 = [0, 1, 0, 1, 1, 1, 1, 0],
α35 = [0, 0, 1, 1, 1, 1, 1, 0],
α36 = [0, 0, 0, 1, 1, 1, 1, 1],
α37 = [1, 1, 1, 2, 1, 0, 0, 0],
α38 = [1, 1, 1, 1, 1, 1, 0, 0],
α39 = [1, 0, 1, 1, 1, 1, 1, 0],
α40 = [0, 1, 1, 2, 1, 1, 0, 0],
α41 = [0, 1, 1, 1, 1, 1, 1, 0],
α42 = [0, 1, 0, 1, 1, 1, 1, 1],
α43 = [0, 0, 1, 1, 1, 1, 1, 1],
α44 = [1, 1, 2, 2, 1, 0, 0, 0],
α45 = [1, 1, 1, 2, 1, 1, 0, 0],
α46 = [1, 1, 1, 1, 1, 1, 1, 0],
α47 = [1, 0, 1, 1, 1, 1, 1, 1],
α48 = [0, 1, 1, 2, 2, 1, 0, 0],
α49 = [0, 1, 1, 2, 1, 1, 1, 0],
α50 = [0, 1, 1, 1, 1, 1, 1, 1],
α51 = [1, 1, 2, 2, 1, 1, 0, 0],
α52 = [1, 1, 1, 2, 2, 1, 0, 0],
α53 = [1, 1, 1, 2, 1, 1, 1, 0],
α54 = [1, 1, 1, 1, 1, 1, 1, 1],
α55 = [0, 1, 1, 2, 2, 1, 1, 0],
α56 = [0, 1, 1, 2, 1, 1, 1, 1],
α57 = [1, 1, 2, 2, 2, 1, 0, 0],
α58 = [1, 1, 2, 2, 1, 1, 1, 0],
α59 = [1, 1, 1, 2, 2, 1, 1, 0],
α60 = [1, 1, 1, 2, 1, 1, 1, 1],
α61 = [0, 1, 1, 2, 2, 2, 1, 0],
α62 = [0, 1, 1, 2, 2, 1, 1, 1],
α63 = [1, 1, 2, 3, 2, 1, 0, 0],
α64 = [1, 1, 2, 2, 2, 1, 1, 0],
α65 = [1, 1, 2, 2, 1, 1, 1, 1],
α66 = [1, 1, 1, 2, 2, 2, 1, 0],
α67 = [1, 1, 1, 2, 2, 1, 1, 1],
α68 = [0, 1, 1, 2, 2, 2, 1, 1],
α69 = [1, 2, 2, 3, 2, 1, 0, 0],
α70 = [1, 1, 2, 3, 2, 1, 1, 0],
α71 = [1, 1, 2, 2, 2, 2, 1, 0],
α72 = [1, 1, 2, 2, 2, 1, 1, 1],
α73 = [1, 1, 1, 2, 2, 2, 1, 1],
α74 = [0, 1, 1, 2, 2, 2, 2, 1],
α75 = [1, 2, 2, 3, 2, 1, 1, 0],
α76 = [1, 1, 2, 3, 2, 2, 1, 0],
α77 = [1, 1, 2, 3, 2, 1, 1, 1],
α78 = [1, 1, 2, 2, 2, 2, 1, 1],
α79 = [1, 1, 1, 2, 2, 2, 2, 1],
α80 = [1, 2, 2, 3, 2, 2, 1, 0],
α81 = [1, 2, 2, 3, 2, 1, 1, 1],
α82 = [1, 1, 2, 3, 3, 2, 1, 0],
α83 = [1, 1, 2, 3, 2, 2, 1, 1],
α84 = [1, 1, 2, 2, 2, 2, 2, 1],
The GraviGUT Algebra Is not a Subalgebra of E8
9
α85 = [1, 2, 2, 3, 3, 2, 1, 0],
α86 = [1, 2, 2, 3, 2, 2, 1, 1],
α87 = [1, 1, 2, 3, 3, 2, 1, 1],
α88 = [1, 1, 2, 3, 2, 2, 2, 1],
α89 = [1, 2, 2, 4, 3, 2, 1, 0],
α90 = [1, 2, 2, 3, 3, 2, 1, 1],
α91 = [1, 2, 2, 3, 2, 2, 2, 1],
α92 = [1, 1, 2, 3, 3, 2, 2, 1],
α93 = [1, 2, 3, 4, 3, 2, 1, 0],
α94 = [1, 2, 2, 4, 3, 2, 1, 1],
α95 = [1, 2, 2, 3, 3, 2, 2, 1],
α96 = [1, 1, 2, 3, 3, 3, 2, 1],
α97 = [2, 2, 3, 4, 3, 2, 1, 0],
α98 = [1, 2, 3, 4, 3, 2, 1, 1],
α99 = [1, 2, 2, 4, 3, 2, 2, 1],
α100 = [1, 2, 2, 3, 3, 3, 2, 1],
α101 = [2, 2, 3, 4, 3, 2, 1, 1],
α102 = [1, 2, 3, 4, 3, 2, 2, 1],
α103 = [1, 2, 2, 4, 3, 3, 2, 1],
α104 = [2, 2, 3, 4, 3, 2, 2, 1],
α105 = [1, 2, 3, 4, 3, 3, 2, 1],
α106 = [1, 2, 2, 4, 4, 3, 2, 1],
α107 = [2, 2, 3, 4, 3, 3, 2, 1],
α108 = [1, 2, 3, 4, 4, 3, 2, 1],
α109 = [2, 2, 3, 4, 4, 3, 2, 1],
α110 = [1, 2, 3, 5, 4, 3, 2, 1],
α111 = [2, 2, 3, 5, 4, 3, 2, 1],
α112 = [1, 3, 3, 5, 4, 3, 2, 1],
α113 = [2, 3, 3, 5, 4, 3, 2, 1],
α114 = [2, 2, 4, 5, 4, 3, 2, 1],
α115 = [2, 3, 4, 5, 4, 3, 2, 1],
α116 = [2, 3, 4, 6, 4, 3, 2, 1],
α117 = [2, 3, 4, 6, 5, 3, 2, 1],
α118 = [2, 3, 4, 6, 5, 4, 2, 1],
α119 = [2, 3, 4, 6, 5, 4, 3, 1],
α120 = [2, 3, 4, 6, 5, 4, 3, 2].
Let Xαi = Xi , and Yαi = Yi be a choice of positive (resp. negative) root vector corresponding
to the root αi .
A basis of V (λ1 ) = [X120 ]so(14)C is given by the 14 positive root vectors:
X97 ,
X101 ,
X104 ,
X107 ,
X109 ,
X111 ,
X113 ,
X114 ,
X115 ,
X116 ,
X117 ,
X118 ,
X119 ,
X120 .
A basis of V (λ1 ) = [Y97 ]so(14)C is given by the 14 negative root vectors:
Y97 ,
Y101 ,
Y104 ,
Y107 ,
Y109 ,
Y111 ,
Y113 ,
Y114 ,
Y115 ,
Y116 ,
Y117 ,
Y118 ,
Y119 ,
Y120 .
A basis of V (λ7 ) = [X112 ]so(14)C is given by the 64 positive root vectors:
X1 ,
X9 ,
X16 ,
X23 ,
X24 ,
X30 ,
X31 ,
X37 ,
X38 ,
X39 ,
X44 ,
X45 ,
X46 ,
X47 ,
X51 ,
X52 ,
X53 ,
X54 ,
X57 ,
X58 ,
X59 ,
X60 ,
X63 ,
X64 ,
X65 ,
X66 ,
X67 ,
X69 ,
X70 ,
X71 ,
X72 ,
X73 ,
X75 ,
X76 ,
X77 ,
X78 ,
X79 ,
X80 ,
X81 ,
X82 ,
X83 ,
X84 ,
X85 ,
X86 ,
X87 ,
X88 ,
X89 ,
X90 ,
X91 ,
X92 ,
X93 ,
X94 ,
X95 ,
X96 ,
X98 ,
X99 ,
X100 ,
X102 ,
X103 ,
X105 ,
X106 ,
X108 ,
X110 ,
X112 .
A basis of V (λ6 ) = [Y1 ]so(14)C is given by the 64 negative root vectors:
Y1 ,
Y9 ,
Y16 ,
Y23 ,
Y24 ,
Y30 ,
Y31 ,
Y37 ,
Y38 ,
Y39 ,
Y44 ,
Y45 ,
Y46 ,
Y47 ,
Y51 ,
Y52 ,
Y53 ,
Y54 ,
10
A. Douglas and J. Repka
Y57 ,
Y58 ,
Y59 ,
Y60 ,
Y63 ,
Y64 ,
Y65 ,
Y66 ,
Y67 ,
Y69 ,
Y70 ,
Y71 ,
Y72 ,
Y73 ,
Y75 ,
Y76 ,
Y77 ,
Y78 ,
Y79 ,
Y80 ,
Y81 ,
Y82 ,
Y83 ,
Y84 ,
Y85 ,
Y86 ,
Y87 ,
Y88 ,
Y89 ,
Y90 ,
Y91 ,
Y92 ,
Y93 ,
Y94 ,
Y95 ,
Y96 ,
Y98 ,
Y99 ,
Y100 ,
Y102 ,
Y103 ,
Y105 ,
Y106 ,
Y108 ,
Y110 ,
Y112 .
Note that ϕ(so(14)C ) = [X74 ]so(14)C . We describe bases of ϕ(so(14)C+ ) and ϕ(so(14)C− ),
respectively:
X2 ,
X3 ,
X4 ,
X5 ,
X6 ,
X7 ,
X8 ,
X10 ,
X11 ,
X12 ,
X13 ,
X14 ,
X15 ,
X17 ,
X18 ,
X19 ,
X20 ,
X21 ,
X22 ,
X25 ,
X26 ,
X27 ,
X28 ,
X29 ,
X32 ,
X33 ,
X34 ,
X35 ,
X36 ,
X40 ,
X41 ,
X42 ,
X43 ,
X48 ,
X49 ,
X50 ,
X55 ,
X56 ,
X61 ,
X62 ,
X68 ,
X74 ;
Y2 ,
Y3 ,
Y4 ,
Y5 ,
Y6 ,
Y7 ,
Y8 ,
Y10 ,
Y11 ,
Y12 ,
Y13 ,
Y14 ,
Y15 ,
Y17 ,
Y18 ,
Y19 ,
Y20 ,
Y21 ,
Y22 ,
Y25 ,
Y26 ,
Y27 ,
Y28 ,
Y29 ,
Y32 ,
Y33 ,
Y34 ,
Y35 ,
Y36 ,
Y40 ,
Y41 ,
Y42 ,
Y43 ,
Y48 ,
Y49 ,
Y50 ,
Y55 ,
Y56 ,
Y61 ,
Y62 ,
Y68 ,
Y74 .
Acknowledgements. The work of A.D. is partially supported by a research grant from the
Professional Staff Congress/ City University of New York (PSC/CUNY). The work of J.R. is
partially supported by the Natural Sciences and Engineering Research Council (NSERC). The
authors would also like to thank the anonymous referees for valuable comments.
References
[1] Baez J., The octonions, Bull. Amer. Math. Soc. 39 (2002), 145–205, math.RA/0105155.
[2] Baez J., Huerta J., The algebra of grand unified theories, Bull. Amer. Math. Soc. 47 (2010), 483–552,
arXiv:0904.1556.
[3] Distler J., Garibaldi S., There is no “theory of everything” inside E8 , Comm. Math. Phys. 298 (2010),
419–436, arXiv:0905.2658.
[4] Douglas A., Kahrobaei D., Repka J., Classification of embeddings of abelian extensions of Dn into En+1 ,
J. Pure Appl. Algebra 217 (2013), 1942–1954.
[5] GAP-Groups, Algorithms, and programming, Version 4.2, 2000, http://www-gap.dcs.st-and.ac.uk/~gap.
[6] Humphreys J.E., Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics,
Vol. 9, Springer-Verlag, New York – Berlin, 1972.
[7] Lisi A.G., An exceptionally simple theory of everything, arXiv:0711.0770.
[8] Lisi A.G., An explicit embedding of gravity and the Standard Model in E8 , in Representation Theory
and Mathematical Physics, Contemp. Math., Vol. 557, Amer. Math. Soc., Providence, RI, 2011, 231–244,
arXiv:1006.4908.
[9] Malcev A.I., Commutative subalgebras of semi-simple Lie algebras, Amer. Math. Soc. Translation 1951
(1951), no. 40, 15 pages.
[10] Minchenko A.N., Semisimple subalgebras of exceptional Lie algebras, Trans. Moscow Math. Soc. 67 (2006),
225–259.
[11] Mkrtchyan R.L., On the map of Vogel’s plane, arXiv:1209.5709.
[12] Nesti F., Percacci R., Chirality in unified theories of gravity, Phys. Rev. D 81 (2010), 025010, 7 pages,
arXiv:0909.4537.
[13] Vogel P., Algebraic structures on modules of diagrams, J. Pure Appl. Algebra 215 (2011), 1292–1339.